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Weighted function spaces: convolutors, multipliers, and mollifiers

Lenny Neyt, Yoshihiro Sawano

TL;DR

This work develops a mollification-based framework to classify Gelfand–Shilov-type spaces defined via a solid translation-invariant Banach function space $E$ and a weight system $\mathcal{W}$. It introduces the moment-wise decomposition factorization property (MDFP) for pairs of mollifying windows and proves complete characterizations of the spaces and their convolutor and multiplier spaces through mollification, including two-window variants that better suppress low-frequency components. The paper establishes independence results for $E^\infty_{\mathcal{W}}$ under mild weight conditions and provides convolution and multiplication mappings that link these spaces to classical ones, such as Schwartz and Hasumi–Silva, thereby unifying several smooth function spaces under a single mollification-driven theory. The results have implications for PDEs and harmonic analysis by giving precise global-decay/localization control and offering tools to analyze generalized functions via mollified representations.

Abstract

We study smooth function spaces of Gelfand-Shilov type, with global behavior governed through a translation-invariant Banach function space and localized via a weight function system. We clarify the roles of the translation-invariant Banach function space, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces-as well as the corresponding convolutor and multiplier spaces-through mollification. For this purpose, we introduce the moment-wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.

Weighted function spaces: convolutors, multipliers, and mollifiers

TL;DR

This work develops a mollification-based framework to classify Gelfand–Shilov-type spaces defined via a solid translation-invariant Banach function space and a weight system . It introduces the moment-wise decomposition factorization property (MDFP) for pairs of mollifying windows and proves complete characterizations of the spaces and their convolutor and multiplier spaces through mollification, including two-window variants that better suppress low-frequency components. The paper establishes independence results for under mild weight conditions and provides convolution and multiplication mappings that link these spaces to classical ones, such as Schwartz and Hasumi–Silva, thereby unifying several smooth function spaces under a single mollification-driven theory. The results have implications for PDEs and harmonic analysis by giving precise global-decay/localization control and offering tools to analyze generalized functions via mollified representations.

Abstract

We study smooth function spaces of Gelfand-Shilov type, with global behavior governed through a translation-invariant Banach function space and localized via a weight function system. We clarify the roles of the translation-invariant Banach function space, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces-as well as the corresponding convolutor and multiplier spaces-through mollification. For this purpose, we introduce the moment-wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.
Paper Structure (14 sections, 41 theorems, 200 equations)

This paper contains 14 sections, 41 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Translation-invariant Banach function spaces
  3. Examples of translation-invariant Banach function spaces
  4. Essential part and module completion
  5. Weight Function Systems
  6. Spaces of Gelfand-Shilov-type
  7. The role of the solid TIBF of bounded type
  8. Convolution
  9. The convolutor and multiplier spaces
  10. The convolutor spaces
  11. The multiplier spaces
  12. Characterizations via mollification
  13. Moment-wise decomposition factorization property
  14. Gelfand-Shilov-type spaces

Key Result

Theorem 1.1

Let $p \in [1, \infty]$. Let $\omega : \mathbb{R}^d \to [0, \infty)$ be a non-decreasing, continuous function such that $\omega$ is moderate, i.e., Let $(\chi^0, \chi^1) \in \mathcal{D} \times \mathcal{D}$ satisfy the MDFP. For any $f \in \mathcal{D}^\prime$, we have that:

Theorems & Definitions (98)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5: Morrey spaces
  • Example 2.6: Wiener amalgam spaces
  • Lemma 2.7
  • proof
  • Lemma 2.8: F-G-BanachSpIntGroupRepAtomicDecomp
  • ...and 88 more